What is the way to reconcile the desire to honor one's parents, while taking into account the father's health condition, appreciating the mother's sacrifices, and avoiding her anger when visiting the father, especially given the difference in the mother's treatment of the brother?

The mother has done well by patiently raising her children, and she will be rewarded. However, the father's frugality does not negate his right to dutifulness, and the mother's right is greater. If the mother prevents visits to the father, he can be visited secretly, and permissible stratagems can be used to combine dutifulness to both of them. The mother's differential treatment does not diminish her right, and you should advise her gently and pray for her.